A Geometry Characteristic for Banach Space with $c^1$-Norm

Abstract

Let $E$ be a Banach space with the $c^1$-norm $\Vert \cdot \Vert$ in $E\backslash \{0\}$ and $S(E)=\{e\in E:\Vert e\Vert =1\}.$ In this paper, a geometry characteristic for $E$ is presented by using a geometrical construct of $S(E).$ That is, the following theorem holds : the norm of $E$ is of $c^1$ in $E\backslash \{0\}$ if and only if $S(E)$ is a $c^1$-submanifold of $E,$ with $\mathrm{codim}S(E)=1.$ The theorem is very clear, however, its proof is non-trivial, which shows an intrinsic connection between the continuous differentiability of the norm $\Vert \cdot \Vert$ in $E\backslash \{0\}$ and differential structure of $S(E).$